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AdS_5 Solutions of Type IIB Supergravity and Generalized Complex Geometry

Maxime Gabella, Jerome P. Gauntlett, Eran Palti, James Sparks, Daniel Waldram

TL;DR

The paper develops a generalized geometric framework for supersymmetric AdS_5 solutions of type IIB supergravity beyond Sasaki–Einstein, treating the six-dimensional cone X as a generalized Calabi–Yau with two compatible pure spinors Ω_±. It identifies generalized holomorphic dilatation and R-symmetry vectors, and performs a generalized reduction to a four-dimensional transversal geometry that inherits a generalized Hermitian structure. By exploiting the symplectic structure on X (under nonzero F_5) it derives Duistermaat–Heckman type integrals for the central charge and for the conformal dimensions of BPS wrapped D3-branes, with localization available in toric cases; the Pilch–Warner solution serves as a detailed example. The results extend known Sasaki–Einstein geometry to a broader class of AdS_5 backgrounds, offering a geometric route to compute SCFT data and guiding construction of new solutions and potential connections to a-maximization and mesonic/mesonic operator spectra.

Abstract

We use the formalism of generalized geometry to study the generic supersymmetric AdS_5 solutions of type IIB supergravity that are dual to N=1 superconformal field theories (SCFTs) in d=4. Such solutions have an associated six-dimensional generalized complex cone geometry that is an extension of Calabi-Yau cone geometry. We identify generalized vector fields dual to the dilatation and R-symmetry of the dual SCFT and show that they are generalized holomorphic on the cone. We carry out a generalized reduction of the cone to a transverse four-dimensional space and show that this is also a generalized complex geometry, which is an extension of Kahler-Einstein geometry. Remarkably, provided the five-form flux is non-vanishing, the cone is symplectic. The symplectic structure can be used to obtain Duistermaat-Heckman type integrals for the central charge of the dual SCFT and the conformal dimensions of operators dual to BPS wrapped D3-branes. We illustrate these results using the Pilch-Warner solution.

AdS_5 Solutions of Type IIB Supergravity and Generalized Complex Geometry

TL;DR

The paper develops a generalized geometric framework for supersymmetric AdS_5 solutions of type IIB supergravity beyond Sasaki–Einstein, treating the six-dimensional cone X as a generalized Calabi–Yau with two compatible pure spinors Ω_±. It identifies generalized holomorphic dilatation and R-symmetry vectors, and performs a generalized reduction to a four-dimensional transversal geometry that inherits a generalized Hermitian structure. By exploiting the symplectic structure on X (under nonzero F_5) it derives Duistermaat–Heckman type integrals for the central charge and for the conformal dimensions of BPS wrapped D3-branes, with localization available in toric cases; the Pilch–Warner solution serves as a detailed example. The results extend known Sasaki–Einstein geometry to a broader class of AdS_5 backgrounds, offering a geometric route to compute SCFT data and guiding construction of new solutions and potential connections to a-maximization and mesonic/mesonic operator spectra.

Abstract

We use the formalism of generalized geometry to study the generic supersymmetric AdS_5 solutions of type IIB supergravity that are dual to N=1 superconformal field theories (SCFTs) in d=4. Such solutions have an associated six-dimensional generalized complex cone geometry that is an extension of Calabi-Yau cone geometry. We identify generalized vector fields dual to the dilatation and R-symmetry of the dual SCFT and show that they are generalized holomorphic on the cone. We carry out a generalized reduction of the cone to a transverse four-dimensional space and show that this is also a generalized complex geometry, which is an extension of Kahler-Einstein geometry. Remarkably, provided the five-form flux is non-vanishing, the cone is symplectic. The symplectic structure can be used to obtain Duistermaat-Heckman type integrals for the central charge of the dual SCFT and the conformal dimensions of operators dual to BPS wrapped D3-branes. We illustrate these results using the Pilch-Warner solution.

Paper Structure

This paper contains 30 sections, 1 theorem, 253 equations.

Table of Contents

  1. Introduction
  2. Generalized geometry
  3. The generalized tangent and spinor bundles
  4. Generalized metrics and complex structures
  5. $AdS_5$ backgrounds as generalized complex geometries
  6. Supersymmetric $AdS_5$ backgrounds
  7. Reformulation as generalized complex geometries
  8. Canonical vector fields
  9. Dilatation symmetry
  10. $R$-symmetry
  11. Generalized reduction of $AdS_5$ backgrounds
  12. Generalized reductions
  13. Generalized reduction from $\xi$
  14. $\mathfrak{g}=\mathbb{R}$ reduction
  15. $\mathfrak{g}=\mathbb{C}$ reduction
  16. ...and 15 more sections

Key Result

Theorem 1

Let $E$ be a generalized tangent space over $M$ with reduction data $(\tilde{\psi},\mathfrak{h},\mu)$. Suppose $E$ is equipped with a $G$-invariant generalized Hermitian structure $(\mathcal{J},G)$. If over $\mu^{-1}(0)$, $\mathcal{J}K^G = K^G$, or equivalently $\mathcal{J}E_K = E_K$, then $\mathcal

Theorems & Definitions (1)

  • Theorem 1: Generalized Hermitian reduction bcg